Advertisements
Advertisements
Question
Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx
Advertisements
Solution
Let I = `int_0^(2π) (1)/(1 + e^(sin x)`dx ...(i)
Applying property,
`int_0^af(x)dx = int_0^af(a-x)dx,` we get
I = `int_0^(2pi) dx/(1+e^(sin(2pi-x)))`
= `int_0^(2pi)dx/(1+e^(-sinx))`
= `int_0^(2pi)dx/(1+1/e^(sinx))`
= `int_0^(2pi)(e^(sinx)dx)/(e^(sinx)+1)` ...(ii)
On adding equations (i) and (ii), we get
2I = `int_0^(2pi)dx/(1+e^(sinx))+int_0^(2pi)(e^(sinx)dx)/(1+e^(sinx))`
= `int_0^(2pi)((1+e^(sinx))/(1+e^(sinx)))dx`
= `int_0^(2pi)1.dx`
⇒ 2I = `[x]_0^(2pi)`
⇒ 2I = [2π]
⇒ I = π
RELATED QUESTIONS
By using the properties of the definite integral, evaluate the integral:
`int_0^2 xsqrt(2 -x)dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^pi log(1+ cos x) dx`
By using the properties of the definite integral, evaluate the integral:
`int_0^4 |x - 1| dx`
The value of `int_0^(pi/2) log ((4+ 3sinx)/(4+3cosx))` dx is ______.
\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.
Evaluate : `int "x"^2/("x"^4 + 5"x"^2 + 6) "dx"`
Evaluate the following integrals : `int_2^5 sqrt(x)/(sqrt(x) + sqrt(7 - x))*dx`
`int_"a"^"b" "f"(x) "d"x` = ______
State whether the following statement is True or False:
`int_(-5)^5 x/(x^2 + 7) "d"x` = 10
Evaluate `int_1^3 x^2*log x "d"x`
The value of `int_-3^3 ("a"x^5 + "b"x^3 + "c"x + "k")"dx"`, where a, b, c, k are constants, depends only on ______.
`int_0^(pi/2) sqrt(cos theta) * sin^2 theta "d" theta` = ______.
If f(x) = |x - 2|, then `int_-2^3 f(x) dx` is ______
The value of `int_1^3 dx/(x(1 + x^2))` is ______
`int_0^1 "dx"/(sqrt(1 + x) - sqrtx)` = ?
`int_("a" + "c")^("b" + "c") "f"(x) "d"x` is equal to ______.
If `f(a + b - x) = f(x)`, then `int_0^b x f(x) dx` is equal to
Evaluate: `int_0^(π/2) 1/(1 + (tanx)^(2/3)) dx`
If `intxf(x)dx = (f(x))/2` then f(x) = ex.
If `int_(-a)^a(|x| + |x - 2|)dx` = 22, (a > 2) and [x] denotes the greatest integer ≤ x, then `int_a^(-a)(x + [x])dx` is equal to ______.
The value of the integral `int_(-1)^1log_e(sqrt(1 - x) + sqrt(1 + x))dx` is equal to ______.
The value of `int_((-1)/sqrt(2))^(1/sqrt(2)) (((x + 1)/(x - 1))^2 + ((x - 1)/(x + 1))^2 - 2)^(1/2)`dx is ______.
What is `int_0^(π/2)` sin 2x ℓ n (cot x) dx equal to ?
`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.
Evaluate `int_1^2(x+3)/(x(x+2)) dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
Solve the following.
`int_0^1e^(x^2)x^3dx`
\[\int_{-2}^{2}\left|x^{2}-x-2\right|\mathrm{d}x=\]
Which property is represented by \[\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(t)\,dt\]?
If \[f(-x)=-f(x)\], what is \[\int_{-a}^{a} f(x)\,dx\]?
